Finite Group Databasebeta

← #15874224

Subgroups — #15874224

Order 3328.

920 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718711C211yesmaximal, characteristic
2#13117420C211yesmaximal, characteristic
3#13115706C211yesmaximal, characteristic, complemented
4#13115696C211yesmaximal, characteristic, complemented
5#13115525C211yesmaximal, characteristic, complemented
6#13115511C211yesmaximal, characteristic, complemented
7#13115477C211yesmaximal, characteristic, complemented
8#13115470C211yesmaximal, characteristic, complemented
9#13115177C211yesmaximal, characteristic, complemented
10#13115177C211yesmaximal, characteristic, complemented
11#13115149C211yesmaximal, characteristic
12#13115033C211yesmaximal, characteristic
13#13114770C211yesmaximal, characteristic
14#13114602C211yesmaximal, characteristic
15C13×#2323C211yesmaximal, characteristic
16C23×Dic26C2211yescharacteristic
17C2×C13×C22⋊Q8C2211yescharacteristic
18C2×C4×Q8×C13C2211yescharacteristic
19C2×C13×#117C2211yescharacteristic
20C22×#6107C2211yescharacteristic
21C22×#6107C2211yescharacteristic
22C2×#88180C2211yescharacteristic
23C13×#391C2211yescharacteristic
24C13×#389—21no—
25C13×#389C2211yescharacteristic
26C13×#388—21no—
27C13×#388—21no—
28C13×#387—21no—
29C13×#380C2211yescharacteristic
30C4×C13×C4⋊C4C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×C22⋊#1055C2211yescharacteristic
34C2×C22⋊#1055C2211yescharacteristic
35#11718711C418yescomplemented
36#11718635—21no—
37#11718635—21no—
38C2×#88050C2211yescharacteristic
39C2×#88050C2211yescharacteristic
40C2×#88050C2211yescharacteristic
41C2×#88050C2211yescharacteristic
42C2×#88050C2211yescharacteristic
43C2×#88050C2211yescharacteristic
44C2×#88048C2211yescharacteristic
45C2×#88047C2211yescharacteristic
46#11718604—21no—
47#11718604—21no—
48#11718590C2211yescharacteristic
49#11718590C2211yescharacteristic
50#11718590C2211yescharacteristic
51#11718590C2211yescharacteristic
52C2×#88043C2211yescharacteristic
53C2×C4×#1055C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718116C2211yescharacteristic
59#11718116C2211yescharacteristic
60C22×Q8×C13C2311yescharacteristic
61C23×C4×C13C2311yescharacteristic
62C23×Dic13—21no—
63C22×Dic26—21no—
64C22×Dic26—22no—
65C22×Dic26—41no—
66C22×Dic26—41no—
67C22×Dic26—41no—
68C22×Dic26—41no—
69C22×Dic26C2×C414yes—
70C22×Dic26C2311yescharacteristic
71C22×Dic26C2311yescharacteristic
72C13×C22⋊Q8—22no—
73C13×C22⋊Q8C2×C414yes—
74C4×Q8×C13—41no—
75C4×Q8×C13—41no—
76C13×#117—41no—
77C13×#117—41no—
78C2×C13×C4⋊C4—21no—
79C2×C13×C4⋊C4—21no—
80C2×C13×C4⋊C4—21no—
81C2×C13×C4⋊C4D411yescharacteristic
82C2×C13×C4⋊C4D411yescharacteristic
83C2×C13×C4⋊C4C2311yescharacteristic
84C2×C13×C4⋊C4C2311yescharacteristic
85C2×C13×C4⋊C4C2311yescharacteristic
86C2×C13×C22⋊C4—21no—
87C2×C13×C22⋊C4—21no—
88C2×C13×C22⋊C4D411yescharacteristic
89C2×C13×C22⋊C4D411yescharacteristic
90C2×C42×C13—21no—
91C2×C42×C13—21no—
92C2×C42×C13C2311yescharacteristic
93C2×C42×C13C2311yescharacteristic
94C2×C22⋊Dic13—21no—
95C2×#6107—41no—
96C2×#6107—41no—
97C2×#6107—41no—
98C2×#6107—41no—
99C2×#6107C2×C414yes—
100C2×#6107C2×C414yes—
101#88180—41no—
102#88180—41no—
103C2×C4⋊Dic13C2311yescharacteristic
104C2×C4⋊Dic13C2311yescharacteristic
105C2×C4×Dic13—21no—
106C22×#1055C2311yescharacteristic
107C22×#1055C2311yescharacteristic
108C22×#1055C2311yescharacteristic
109C22×#1055C2311yescharacteristic
110C13×#95—21no—
111C13×#95—21no—
112C13×#95—21no—
113C13×#95—21no—
114C13×#95—21no—
115#88075—41no—
116#88075—41no—
117#88071—21no—
118#88071—21no—
119#88071—21no—
120#88071—21no—
121#88071—21no—
122#88071—21no—
123#88071—21no—
124#88071—21no—
125C22⋊#1055—22no—
126C22⋊#1055C2×C414yes—
127#88050—41no—
128#88050—41no—
129#88050—41no—
130#88050—41no—
131#88050C2×C414yes—
132#88050C2×C414yes—
133#88048—41no—
134#88047—41no—
135#88043—41no—
136C4×#1055—41no—
137#22054—131nomaximal, Sylow
138C24×C13C2×D411yescharacteristic
139C2×Q8×C13—21no—
140C2×Q8×C13—21no—
141C2×Q8×C13—21no—
142C2×Q8×C13—21no—
143C2×Q8×C13—21no—
144C2×Q8×C13C22×C412yes—
145C22×C4×C13—21no—
146C22×C4×C13—21no—
147C22×C4×C13—21no—
148C22×C4×C13—21no—
149C22×C4×C13—21no—
150C22×C4×C13—21no—
151C22×C4×C13—21no—
152C22×C4×C13—21no—
153C22×C4×C13C22×C412yes—
154C22×C4×C13C2×D411yescharacteristic
155C22×C4×C13C2×D411yescharacteristic
156C22×C4×C13C2×D411yescharacteristic
157C22×C4×C13C2×D411yescharacteristic
158C22×C4×C13C2×D411yescharacteristic
159C22×C4×C13#4112yes—
160C22×C4×C13C2411yescharacteristic
161C22×Dic13—24no—
162C22×Dic13—21no—
163C22×Dic13—21no—
164C22×Dic13—21no—
165C22×Dic13—21no—
166C22×Dic13—42no—
167C22×Dic13—42no—
168C2×Dic26—21no—
169C2×Dic26—21no—
170C2×Dic26—21no—
171C2×Dic26—21no—
172C2×Dic26—21no—
173C2×Dic26—21no—
174C2×Dic26—21no—
175C2×Dic26—21no—
176C2×Dic26—24no—
177C2×Dic26—21no—
178C2×Dic26—21no—
179C2×Dic26—44no—
180C2×Dic26—44no—
181C2×Dic26—42no—
182C2×Dic26—42no—
183C2×Dic26—44no—
184C2×Dic26—44no—
185C2×Dic26C22×C412yes—
186C2×Dic26C22×C412yes—
187C13×C4⋊C4—24no—
188C13×C4⋊C4—21no—
189C13×C4⋊C4—21no—
190C13×C4⋊C4—21no—
191C13×C4⋊C4—21no—
192C13×C4⋊C4—21no—
193C13×C4⋊C4—41no—
194C13×C4⋊C4—41no—
195C13×C4⋊C4—42no—
196C13×C4⋊C4—81no—
197C13×C4⋊C4C22×C412yes—
198C13×C22⋊C4—24no—
199C13×C22⋊C4—41no—
200C13×C22⋊C4—41no—
201C13×C22⋊C4—81no—
202C42×C13—21no—
203C42×C13—21no—
204C42×C13—21no—
205C42×C13—21no—
206C42×C13—42no—
207C42×C13—81no—
208C22⋊Dic13—42no—
209#6107—44no—
210#6107—44no—
211C4⋊Dic13—21no—
212C4⋊Dic13—21no—
213C4⋊Dic13—21no—
214C4⋊Dic13—21no—
215C4×Dic13—42no—
216C2×#1055—21no—
217C2×#1055—21no—
218C2×#1055—21no—
219C2×#1055—21no—
220C2×#1055—21no—
221C2×#1055—21no—
222C2×#1055—21no—
223C2×#1055—21no—
224C2×#1055—21no—
225C2×#1055—21no—
226C2×#1055C22×C412yes—
227C2×#1055C22×C412yes—
228C2×#451—131no—
229#2323D1311yescharacteristic
230#2310—131no—
231#1971—131no—
232#1969—131no—
233#1935—131no—
234#1933—131no—
235#1928—131no—
236#1927—131no—
237#1878—131no—
238#1878—131no—
239#1873—131no—
240#1854—131no—
241#1787—131no—
242#1750—131no—
243C23×C13—21no—
244C23×C13—21no—
245C23×C13—21no—
246C23×C13—21no—
247C23×C13—21no—
248C23×C13—21no—
249C23×C13C4×D412yes—
250C23×C13C22×D411yescharacteristic
251Q8×C13—22no—
252Q8×C13—22no—
253Q8×C13—41no—
254Q8×C13—41no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—22no—
258C2×C4×C13—22no—
259C2×C4×C13—22no—
260C2×C4×C13—22no—
261C2×C4×C13—22no—
262C2×C4×C13—22no—
263C2×C4×C13—21no—
264C2×C4×C13—21no—
265C2×C4×C13—21no—
266C2×C4×C13—21no—
267C2×C4×C13—21no—
268C2×C4×C13—21no—
269C2×C4×C13—21no—
270C2×C4×C13—21no—
271C2×C4×C13—21no—
272C2×C4×C13—21no—
273C2×C4×C13—21no—
274C2×C4×C13—21no—
275C2×C4×C13—41no—
276C2×C4×C13—41no—
277C2×C4×C13—41no—
278C2×C4×C13—41no—
279C2×C4×C13—41no—
280C2×C4×C13—41no—
281C2×C4×C13—41no—
282C2×C4×C13—41no—
283C2×C4×C13—41no—
284C2×C4×C13—41no—
285C2×C4×C13C4×D412yes—
286C2×C4×C13C23×C411yescharacteristic
287C2×C4×C13C2×#4111yescharacteristic
288C2×C4×C13H(5,𝔽2)11yescharacteristic
289C2×C4×C13H(5,𝔽2)11yescharacteristic
290C2×C4×C13H(5,𝔽2)11yescharacteristic
291C2×C4×C13H(5,𝔽2)11yescharacteristic
292C2×Dic13—22no—
293C2×Dic13—22no—
294C2×Dic13—48no—
295C2×Dic13—41no—
296C2×Dic13—41no—
297C2×Dic13—41no—
298C2×Dic13—41no—
299C2×Dic13—41no—
300C2×Dic13—41no—
301C2×Dic13—41no—
302C2×Dic13—41no—
303C2×Dic13—41no—
304C2×Dic13—41no—
305Dic26—22no—
306Dic26—22no—
307Dic26—22no—
308Dic26—22no—
309Dic26—48no—
310Dic26—41no—
311Dic26—41no—
312Dic26—41no—
313Dic26—41no—
314#1055—22no—
315#1055—22no—
316#1055—41no—
317#1055—41no—
318C23×Q8—131no—
319C22×Q16—131no—
320C22×Q16—131no—
321C2×C22⋊Q8C2×D1311yescharacteristic
322C2×C4×Q8C2×D1311yescharacteristic
323C2×#117—131no—
324C2×#117C2×D1311yescharacteristic
325#451—138no—
326C2×#107—131no—
327C2×#106—131no—
328C2×#103—131no—
329C2×#103—131no—
330C2×#103—131no—
331C2×#103—131no—
332C2×#103—131no—
333C2×#103—131no—
334C2×#103—131no—
335C2×#103—131no—
336C2×C22⋊C8—131no—
337C2×C22⋊C8—131no—
338C2×#97—131no—
339C2×C4×C8—131no—
340#391—131no—
341#391—131no—
342#391—131no—
343#391—131no—
344#391—261no—
345#391—261no—
346#391C2×D1311yescharacteristic
347#389—21no—
348#389C2×D1311yescharacteristic
349#388—21no—
350#388—21no—
351#387—21no—
352#381—261no—
353#381—261no—
354#380C2×D1311yescharacteristic
355C4×C4⋊C4C2×D1311yescharacteristic
356#340—131no—
357#340—131no—
358#340—131no—
359#340—131no—
360#336—131no—
361#336—131no—
362C22×C13—22no—
363C22×C13—22no—
364C22×C13—22no—
365C22×C13—41no—
366C22×C13—41no—
367C22×C13—41no—
368C22×C13—41no—
369C22×C13C2×C4×D411yescharacteristic
370C22×C13#53411yescharacteristic
371C22×C13#53511yescharacteristic
372C22×C13#57711yescharacteristic
373C22×C13#57811yescharacteristic
374C22×C13#57811yescharacteristic
375C22×C13#57811yescharacteristic
376C4×C13—22no—
377C4×C13—42no—
378C4×C13—41no—
379C4×C13—41no—
380C4×C13—41no—
381C4×C13—41no—
382C4×C13—42no—
383C4×C13—81no—
384C4×C13—81no—
385C4×C13#51811yescharacteristic
386C4×C13#51811yescharacteristic
387C4×C13#53811yescharacteristic
388C4×C13#53811yescharacteristic
389Dic13—42no—
390Dic13—42no—
391Dic13—81no—
392Dic13—81no—
393C22×Q8—134no—
394C22×Q8—131no—
395C22×Q8—131no—
396C22×Q8—262no—
397C22×Q8—261no—
398C22×Q8—521no—
399C22×Q8—521no—
400C22×Q8—521no—
401C22×Q8—521no—
402C22×Q8C22×D1311yescharacteristic
403C23×C4—261no—
404C23×C4C22×D1311yescharacteristic
405C2×Q16—134no—
406C2×Q16—134no—
407C2×Q16—521no—
408C2×Q16—521no—
409C2×Q16—521no—
410C2×Q16—521no—
411C22×C8—131no—
412C22×C8—131no—
413C22×C8—131no—
414C22×C8—131no—
415C22⋊Q8—22no—
416C22⋊Q8C4×D1314yes—
417C4×Q8—41no—
418C4×Q8—41no—
419#117—41no—
420#117—41no—
421#117—521no—
422#117—521no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—131no—
427C2×C4⋊C4—131no—
428C2×C4⋊C4C22⋊D1311yescharacteristic
429C2×C4⋊C4C22⋊D1311yescharacteristic
430C2×C4⋊C4C22×D1311yescharacteristic
431C2×C4⋊C4C22×D1311yescharacteristic
432C2×C4⋊C4C22×D1311yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—261no—
436C2×C22⋊C4C22⋊D1311yescharacteristic
437C2×C22⋊C4C22⋊D1311yescharacteristic
438C2×C42—21no—
439C2×C42—21no—
440C2×C42—261no—
441C2×C42C22×D1311yescharacteristic
442C2×C42C22×D1311yescharacteristic
443#107—521no—
444#106—521no—
445#103—134no—
446#103—134no—
447#103—521no—
448#103—521no—
449#103—521no—
450#103—521no—
451#103—521no—
452#103—521no—
453C22⋊C8—134no—
454C22⋊C8—262no—
455#97—521no—
456C4×C8—521no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—261no—
463#95—261no—
464#95—261no—
465#95—261no—
466#95—261no—
467#95—261no—
468#95—261no—
469#95—261no—
470C2×C13—42no—
471C2×C13#232111yescharacteristic
472C2×C13#294711yescharacteristic
473C2×C13#295011yescharacteristic
474C2×C13#320111yescharacteristic
475C2×C13#320211yescharacteristic
476C2×C13#322311yescharacteristic
477C2×C13#322511yescharacteristic
478C24C2×C22⋊D1311yescharacteristic
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—132no—
485C2×Q8—132no—
486C2×Q8—264no—
487C2×Q8—261no—
488C2×Q8—261no—
489C2×Q8—261no—
490C2×Q8—261no—
491C2×Q8—261no—
492C2×Q8—261no—
493C2×Q8—261no—
494C2×Q8—261no—
495C2×Q8—261no—
496C2×Q8—261no—
497C2×Q8—524no—
498C2×Q8—524no—
499C2×Q8—522no—
500C2×Q8—522no—
501C2×Q8—524no—
502C2×Q8—524no—
503C2×Q8C2×C4×D1312yes—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—264no—
513C22×C4—261no—
514C22×C4—261no—
515C22×C4—261no—
516C22×C4—261no—
517C22×C4—522no—
518C22×C4—522no—
519C22×C4C2×C4×D1312yes—
520C22×C4#612712yes—
521C22×C4C2×C22⋊D1311yescharacteristic
522C22×C4C2×C22⋊D1311yescharacteristic
523C22×C4C2×C22⋊D1311yescharacteristic
524C22×C4C2×C22⋊D1311yescharacteristic
525C22×C4C2×C22⋊D1311yescharacteristic
526C22×C4C23×D1311yescharacteristic
527Q16—524no—
528Q16—524no—
529C2×C8—132no—
530C2×C8—132no—
531C2×C8—261no—
532C2×C8—261no—
533C2×C8—261no—
534C2×C8—261no—
535C2×C8—261no—
536C2×C8—261no—
537C2×C8—261no—
538C2×C8—261no—
539C2×C8—261no—
540C2×C8—261no—
541C4⋊C4—24no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—41no—
548C4⋊C4—41no—
549C4⋊C4—42no—
550C4⋊C4—81no—
551C4⋊C4—261no—
552C4⋊C4—261no—
553C4⋊C4—261no—
554C4⋊C4—261no—
555C4⋊C4C2×C4×D1312yes—
556C22⋊C4—24no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—81no—
560C22⋊C4—522no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—42no—
566C42—81no—
567C42—522no—
568C13#2205411yescharacteristic, complemented, Sylow
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23C4×C22⋊D1312yes—
576C23C22×C22⋊D1311yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581Q8—262no—
582Q8—262no—
583Q8—262no—
584Q8—262no—
585Q8—528no—
586Q8—521no—
587Q8—521no—
588Q8—521no—
589Q8—521no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—262no—
621C2×C4—262no—
622C2×C4—528no—
623C2×C4—521no—
624C2×C4—521no—
625C2×C4—521no—
626C2×C4—521no—
627C2×C4—521no—
628C2×C4—521no—
629C2×C4—521no—
630C2×C4—521no—
631C2×C4—521no—
632C2×C4—521no—
633C2×C4C4×C22⋊D1312yes—
634C2×C4C22×C4×D1311yescharacteristic
635C2×C4C2×#612711yescharacteristic
636C2×C4#8825111yescharacteristic
637C2×C4#8825111yescharacteristic
638C2×C4#8825611yescharacteristic
639C2×C4#8825611yescharacteristic
640C8—262no—
641C8—262no—
642C8—521no—
643C8—521no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C2×C4×C22⋊D1311yescharacteristic
652C22#1171950911yescharacteristic
653C22#1171953511yescharacteristic
654C22#1171954111yescharacteristic
655C22#1171954311yescharacteristic
656C22#1171954311yescharacteristic
657C22#1171954311yescharacteristic
658C4—22no—
659C4—42no—
660C4—41no—
661C4—41no—
662C4—42no—
663C4—41no—
664C4—41no—
665C4—81no—
666C4—81no—
667C4—522no—
668C4—522no—
669C4—1041no—
670C4—1041no—
671C4#1171925011yescharacteristic
672C4#1171925011yescharacteristic
673C4#1171926511yescharacteristic
674C4#1171926511yescharacteristic
675C2—42no—
676C2#1311749011yescharacteristic
677C2#1312367511yescharacteristic
678C2#1312370211yescharacteristic
679C2#1312571211yescharacteristic
680C2#1312572711yescharacteristic
681C2#1312591311yescharacteristic
682C2#1312593911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2323C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#2323D13
hypocenterC13#22054
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #13115706C2semidirect
4. #13115696C2semidirect
5. #13115525C2semidirect
6. #13115511C2semidirect
7. #13115477C2semidirect
8. #13115470C2semidirect
9. #13115177C2semidirect
10. #13115177C2semidirect
35. #11718711C4semidirect
568. C13#22054semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#117187112
#131174202
#131157062
#131156962
#131155252
#131155112
#131154772
#131154702
#131151772
#131151492
#131150332
#131147702
#131146022
C13×#23232
#2205413